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Speaker: Alejandro Morales (Montreal)

Title: The generalized Pitman-Stanley flow polytope

Abstract: In 1999, Pitman and Stanley introduced the polytope bearing their name along with a study of its faces, lattice points, and volume. This polytope is well-studied due to its connections to parking functions, lattice path matroids, generalized permutahedra/polymatroids, and flow polytopes. Its lattice points correspond to plane partitions of skew shape with entries 0 and 1. Pitman and Stanley remarked that their polytope can be generalized so that lattice points correspond to plane partitions of skew shape with entries 0,1,...,m. Since then, this generalization has been untouched. We study this polytope and show that it can also be realized as a flow polytope of a grid graph. In this talk I will discuss characterizations of its vertices and give formulas for the number of vertices and faces as well as old and new formulas for the number of lattice points and volume in terms of rectangular Standard Young Tableaux. The new formulas come from the volume polynomial formulas of flow polytopes in terms of vector partition functions of Baldoni and Vergne and lattice point formulas of Stanley of marked order polytopes.

This is joint work with Maura Hegarty, William Dugan, and Annie Raymond.